aA Guide to FactorialFrom the very humble beginning to the finity.By: Fabi

... somewhere in the heavens ... they are waiting ... they exist in another dimension far removed from our own and our mundane concerns ... they live in the gaping yawn of eternity in the sea of infinity ... few are aware of their existence ... the contemplation of the smaller members of their kind is enough to drive men insane ... they exist in a vast realm of shadowy existence far greater than any human being can imagine ... a realm in fact far far vaster than the known universe! An elect of the smallest of its denizens are known only by arcane signs and sigils, but an infinitely greater multitude are known by no name and no sign and there is no known magic that can summon them! They make the universe tremble in freight. They are the bane of every high, noble, terrifying or transcendent concept of man ... for these shrink into obscurity in their presence and without them nothing so great or terrible can even be imagined or summoned ... what do I speak of?-Sbiis Saibian 1. PrefaceWhen I was in highschool, I have a weird fascination towards big number. At that time, I didn't know any big number, even the "smaller" one like Graham Number, let alone "bigger" one like TREE(3), BB(n), Aeckerman's Arrow Notation, Conway's Arrow Notation, BEAF Notation, fast-growing hierarchy function, etc. Long story short, I found Googology Wikia and I learn so much things there (and of course I didn't understand 99% of what I've read). I also start to binge watching David Metzler's "Large Number" lecture series and learning ordinal number. Since there are few Googology that use factorial-like notation, I have an urge to write a system of notation that employs factorial. I didn't plan this Googology system to be stronger than other noation. Of course not. I can't understand majority of Sbiis Sabian's notation (Extensible-E Notation), let alone to surpass him. I also didn't understand fast growing hierarchy beyond the 𝜀0 level. I write this purely to excerices my understanding at recursive function and diagonalization, and for fun. 2. Table of Contents2.1. Factorial-like notationIn this section, we are dealing with factorial-like notation, a notation that iterates certain operator for n down to the (n-1) and so on, until 3, 2, 1. For example, regular factorial (n!) is no more that a notation that iterates multiplication from n down to the (n-1) and so on, until 3, 2, 1. 2.2. φ Function Family (φ, ϕ, and Φ)How to iterates the number of factorial-like notation after n? For example, we want to write3!!!=((3!)!)!=2.6×101746Take a look that the number of factorial is three. So, we can devise a function that count the number of factorial-like operator after 3. 2.3. Combining The Two: Larger Number Defined2.4. Interseting Gedankenexperiment Big Number